The two questions above can also be asked in the context of a particular class of topological spaces. The ones we have in mind were introduced by Dana Scott, [1]}, [2]}, and are known collectively as domains. The characteristic feature of domains is that they carry a partial order and that their topology is completely determined by the order.
More precisely, let \(P\)  be a poset and \(U\)  a subset of \(P\) . One says that \(U\)  is Scott open, if (i) \(U\)  is an upper set, and (ii) \(U\)  is inaccessible by directed joins. The set \(\sigma (P)\)  of all Scott opens of \(P\)  forms the Scott topology on \(P\) , and \(\Sigma P := (P,\sigma (P))\)  is called the Scott space of \(P\) . In what follows, for a poset \(P\) , we write \(\Gamma (P)\)  to always mean the lattice of Scott-closed subsets of \(P\) .
